Complex Dynamics in Adaptive Networks

The origin of life on earth and its evolution to modern human life required severalfundamental transitions: genes had to form a genome, uni-cellular organismshad to become multi-cellular organisms and humans had to organize in societies.None of these transitions would have been possible without the emergence ofcooperation among the original entities of selection, allowing the formation ofnew, more complex, and independently reproducing, entities.

Interactions that involve cooperation are often formalized in terms games,such as the famous prisoner’s dilemma. Players either cooperate or defect uponinteraction. Cooperators pay a cost (c) to provide a benefit (b > c) to theirpartners. Defectors refuse to cooperate, and therefore incur no costs whilestill ripping the benefits provided by others. The accumulation of receivedbenefits and expended costs defines the reproductive success of an individual.Reproduction is either genetic or cultural, the latter meaning that the behavior ofsuccessful individuals tends to be imitated more often and therefore spreads in thepopulation. When all individuals are equally likely to interact, the mathematicsof cooperation (evolutionary game theory) predicts that defection will prevail.

Cooperative behavior is evolutionary disadvantageous, unless additionalmechanisms foster its emergence. Recent theoretical studies suggest that thestructure of our social network may act as such a mechanism. One of theassumptions of these studies is that individuals adjust their behavior by imitatingsuccessful social acquaintances. Recent experimental work does, however, indicatethat humans often deviate from this learning paradigm: we do not alwaysfollow the example of our peers, but also try to innovate by individuallyexploring alternatives. In the first part of this dissertation, we propose tomodel exploration behavior in terms of an individual-based learning rule. Weassume that individuals adjust their behavior based purely on such an individualbasedlearning scheme and investigate how and to which extent the structureof the social network affects the final game behavior. We show by means ofcomputer simulations that in that case the network structure no longer playssuch a prominent role.

Our results also suggest how the learning dynamics will change when takinginto account that realistic social networks are dynamical entities: we regularlyestablish new contacts while existing contacts may fade away. The lifetime ofan interaction often depends on the behavior of the individuals involved, leadingto a complex interplay between individual behavior and network structure. Thistype of networks are known as adaptive networks: a dynamical process that takesplace on the network, in this case the learning dynamics associated with eachindividual, influences the network structure and vice versa. Recent theoreticalwork indicates that adaptive networks provide an environment that may promotecooperative behavior. This is for instance the case when individuals are given thechance to break unwanted interactions, while keeping the good ones.

The evolutionary mechanism determining individuals’ willingness to changepartner remains unclear, and forms the second part of this dissertation. Toaddress this question, we assign each individual a topological strategy, whichdetermines how he handles unfavorable interactions. Depending on his topologicalstrategy, an individual will either break contact quickly in case he is dissatisfiedabout an interaction, or behave in a more loyal way towards his social partners.In this context, we study the co-evolution of the game strategy, the topologicalstrategy and the network structure.

We show, by means of computer simulations and analytical methods, thatthe selection pressure on the topological strategy of cooperators will not be asstrong as on the topological strategy of defectors. Cooperators are only forced tobecome less loyal when competition is fierce between cooperators and defectors.As a consequence, defectors will exhibit a limited set of topological strategies,whereas cooperators portray diversity in their topological strategy. Here, weshow that cooperation evolves more easily when individuals have the choice froma broad spectrum of topological strategies.

The last part of this dissertation addresses adaptive networks in a completelydifferent context. The network now represents the paths along which a diseasemay spread in a population. Infected individuals may transmit the diseaseto their neighbors in the network. Most models of disease spreading classifyindividuals in different health classes. The simplest model only distinguishesbetween healthy and infected individuals. Depending on the disease at hand, andthe sophistication of the model, additional health classes are included, accountingfor instance for immunity, latency periods, etc. Additional realism can also beadded by taking geographical characteristics, heterogeneity, mobility, etc. intoaccount.

In this dissertation we emphasize an aspect that has been neglected in moststudies: human behavior. Most often, individuals are assumed to behave in thesame way, irrespective of their health status or the number of infected in thepopulation. This is obviously not true in reality. Sick individuals are expectedto travel less than healthy individuals, and will for instance refrain from goingto work. Conversely, healthy individuals may try to escape from infection byavoiding contact with infected. As such, we have another example of an adaptivenetwork. Indeed, the structure of the network is dynamic and coupled to thedynamics that takes place on the network: epidemic spreading. Here, we proposea model for epidemic spreading in an adaptive network, which allows for ananalytical treatment in the limit where the network adjusts faster than the diseasespreads. We use computer simulations to show that the analytical predictions alsohold for a much wider range of scenarios. Furthermore, we demonstrate that inan adaptive network, the infectiousness of a disease depends on the fraction ofinfected and the capacity of healthy individuals to avoid contact with infected.